Dear Martin,
You say,
> I have learned mathematics, a body of knowledge and a methodology.
But you also developed interests, intuitions and habits of thought
and, most important for our purposes, you came to find certain things
obvious because that is how they were presented. All of this is part
of the mindset you say you didn't acquire.
> I have no more acquired a "mindset" than I do when I learn chemistry.
This claim is extremely helpful. We can now focus on something we
all have experienced, learning classical math. Although not under this
description because we were led to believe that we were learning math,
not a kind of math. The unstated belief that classical math = math,
i.e., that LEM is a brute fact about mathematics, is a fundamental part
of the classical mindset.
The claim is also exciting. If you didn't develop a classical
mindset (because you didn't develop any mindset) and, therefore, have
no distinctively classical intuitions, interests or habits of thought
(because you didn't develop any mathematical intuitions, interests or
habits of thought) then, almost surely, neither did I nor anyone else.
In which case, everything that I've been saying for the past 30 years
about shifting between mindsets must be a delusion. I see no other
explanation.
On the other hand, if I'm not deluded about mindsets, then why do
you nevertheless think that you didn't acquire one. Do you suppose
that, if you had, you would necessarily be aware of it? That's not
how the mind works and, in particular, it is not how learning works,
whether it's learning math, chemistry or typing.
Awareness of a mindset that you possess has to be learned, e.g., by
learning to see certain things as manifestations of it. If you're not
aware of having it, that may be a good reason for not assuming that you
do. (Unless you accept on trust that certain things of which you are
aware are manifestations of it, just as we do for our stomach, brain
and other organs to whom we haven't been formally introduced.) But it
is not a good reason for assuming that you haven't acquired the mindset.
So, imagine yourself at your desk. If, as you believe, your only
companions are your methods of proof and body of knowledge, then you
cannot acquire an interest in working on anything. This is because
interests and intuitions are neither methods of proof nor parts of
your body of knowledge, which, according to you, is all that you have.
Finally, for a blast from the classical mindset that is difficult
to mistake for anything else, see Hilbert's remarks in "The fom" about
the evils of math without LEM. In it, he commits a blunder of epic
proportions by observing constructive math only in a classical mindset
when, for his argument about how math should be done, it is essential
that he observe it in a constructive one.
With best regards,
Gabriel